4. Iterative Solvers#
So far we have used direct solvers to solve the linear system of equations. Although a direct solver can profit from the sparse matrix, it’s arithmetic complexity is sub-optimal (i.e. not linear in the number of degrees of freedom). For large-scale problems iterative solvers are a must.
The conjugate gradient (cg) method is the standard method for symmetric and positive definite matrices. It’s convergence rate depends on a preconditioner, what is a cheap approximative inverse to the matrix.
from ngsolve import *
from ngsolve.webgui import Draw
We generate a 3D geometry and mesh using the OCC constructive solid geometry (CSG) modeler:
from netgen.occ import *
cube = Box((0,0,0),(1,1,1))
cyl = Cylinder((0,0.5,0.5),X, r=0.2, h=1)
cube.faces.name = "outer"
cyl.faces.name = "cyl"
shape = cube-cyl
Draw(shape);
ngmesh = OCCGeometry(shape).GenerateMesh(maxh=0.1)
for l in range(1):
ngmesh.Refine()
mesh = Mesh(ngmesh)
mesh.Curve(3)
Draw (mesh);
fes = H1(mesh, order=3, dirichlet="outer", wb_withedges=False)
print ("we have", fes.ndof, "unknowns")
u,v = fes.TnT()
a = BilinearForm(grad(u)*grad(v)*dx)
f = LinearForm(v*dx)
# c = Preconditioner(a, "direct", inverse="sparsecholesky")
c = Preconditioner(a, "local")
# c = Preconditioner(a, "bddc")
gfu = GridFunction(fes)
we have 189222 unknowns
assemble system and setup preconditioner in parallel:
with TaskManager():
a.Assemble()
f.Assemble()
solve the system using the preconditioned conjugate gradient method:
from ngsolve.krylovspace import CGSolver
with TaskManager():
inv = CGSolver(mat=a.mat, pre=c.mat, printrates='\r', maxiter=400)
gfu.vec.data = inv * f.vec
CG iteration 1, residual = 0.02357937034822143
CG iteration 2, residual = 0.036539248269629385
CG iteration 3, residual = 0.04803583710748369
CG iteration 4, residual = 0.04826753355485345
CG iteration 5, residual = 0.03544707042382128
CG iteration 6, residual = 0.02924442241427406
CG iteration 7, residual = 0.030127553489708934
CG iteration 8, residual = 0.0277742921629846
CG iteration 9, residual = 0.026155491369476966
CG iteration 10, residual = 0.02541310684537252
CG iteration 11, residual = 0.023336619745883216
CG iteration 12, residual = 0.020687808569611787
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CG iteration 14, residual = 0.01608336074951512
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CG iteration 216, residual = 2.1104238679326894e-14
CG converged in 216 iterations to residual 2.1104238679326894e-14
Draw (gfu, draw_vol=False);